I'm wondering whether you can tell quickly just from looking at a Lagrangian whether a given point $q^0$ is an equilibrium point. Obviously all you have to do is verify it satisfies the E-L equations, but they are messy in complicated situations. Is there a conceptual shortcut I could take, or am I expecting too much?

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I think you are asking too much. The space of equilibria is some subspace of the space of all configurations. Geometrically, pretty much the best presentation of a submanifold one can give (if all you want to check is whether certain points lie in it) is as the zero set of some functions (in this case the Euler-Lagrange equations).